The only thing which makes complex numbers different from 2-dimensional vectors or any other two-component mathematical object is their multiplication, right? Complex multiplication has uses in rotations but we can easily achieve that using polar co-ordinates. And, their other applications in co-ordinate geometry can be done by replacing them with vectors. And, their remaining use is in quantum mechanics where wave amplitudes are complex. But, do we ever need to multiply these wave amplitudes? I mean, if we don't, then this use of theirs can also be achieved by two numbers separated by commas and enclosed in brackets without ever needing to define square root of negative numbers.